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Answers here have already addressed several important points. To quickly summarize:

  • There is no consistent test that can determine whether a set of data truly follow a distribution or not.
  • Tests are no substitute for visually inspecting the data and models to identify high leverage, high influence observations and commenting on their effects on models.
  • The assumptions for many regression routines are often misquoted as requiring normally distributed "data" [residuals] and that this is interpreted by novice statisticians as requiring that the analyst formally evaluate this in some sense before proceeding with analyses.

I am adding an answer firstly to cite to one of my, personally, most frequently accessed and read statistical articles: "The Importance of Normality Assumptions in Large Public Health Datasets" by Lumley et. al. It is worth reading in entirety. The summary states:

The t-test and least-squares linear regression do not require any assumption of Normal distribution in sufficiently large samples. Previous simulations studies show that “sufficiently large” is often under 100, and even for our extremely non-Normal medical cost data it is less than 500. This means that in public health research, where samples are often substantially larger than this, the t-test and the linear model are useful default tools for analyzing differences and trends in many types of data, not just those with Normal distributions. Formal statistical tests for Normality are especially undesirable as they will have low power in the small samples where the distribution matters and high power only in large samples where the distribution is unimportant.

While the large-sample properties of linear regression are well understood, there has been little research into the sample sizes needed for the Normality assumption to be unimportant. In particular, it is not clear how the necessary sample size depends on the number of predictors in the model.

The focus on Normal distributions can distract from the real assumptions of these methods. Linear regression does assume that the variance of the outcome variable is approximately constant, but the primary restriction on both methods is that they assume that it is sufficient to examine changes in the mean of the outcome variable. If some other summary of the distribution is of greater interest, then the t-test and linear regression may not be appropriate.

To summarize: normality is generally not worth the discussion or the attention it receives in contrast to the importance of answering a particular scientific question. If the desire is to summarize mean differences in data, then the t-test and ANOVA or linear regression are justified in a much broader sense. Tests based on these models remain of the correct alpha level, even when distributional assumptions are not met, although power may be adversely affected.

The reasons why normal distributions may receive the attention they do may be for classical reasons, where exact tests based on F-distributions for ANOVAs and Student-T-distributions for the T-test could be obtained. The truth is, among the many modern advancements of science, we generally deal with larger datasets than were collected previously. If one is in fact dealing with a small dataset, the rationale that those data are normally distributed cannot come from those data themselves: there is simply not enough power. Remarking on other research, replications, or even the biology or science of the measurement process is, in my opinion, a much more justified approach to discussing a possible probability model underlying the observed data.

For this reason, opting for a rank-based test as an alternative misses the point entirely. However, I will agree that using robust variance estimators like the jackknife or bootstrap offer important computational alternatives that permit conducting tests under a variety of more important violations of model specification, such as independence or identical distribution of those errors.

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